Asymptotics of signed Bernoulli convolutions scaled by multinacci numbers

Abstract

We study the signed Bernoulli convolution $${\nu }_{\beta }^{(n)}={\ast }_{j=1}^n\left(\frac{1}{2}{\delta }_{{\beta }^{-j}}-\frac{1}{2}{\delta }_{-{\beta }^{-j}}\right),n\ge 1$$ where $\beta >1$ satisfies $${\beta }^m={\beta }^{m-1}+\cdots +\beta +1$$ for some integer $m\ge 2$. When $m$ is odd, we show that the variation $|{\nu }_{\beta }^{(n)}|$ coincides the unsigned Bernoulli convolution $${\mu }_{\beta }^{(n)}={\ast }_{j=1}^n\left(\frac{1}{2}{\delta }_{{\beta }^{-j}}+\frac{1}{2}{\delta }_{-{\beta }^{-j}}\right).$$ When $m$ is even, we obtain the exact asymptotic of the total variation $\Vert {\nu }_{\beta }^{(n)}\Vert$ as $n\to \infty$.

15 pages